Tidal force and the Roche limit

What just happened?

The moon slipped below the Roche limit. The side of the moon nearer the planet is pulled harder than the far side — this difference in forces is the tidal force. The distance shrank, so the tides surged and overcame the gravity that holds the moon together. The result: disruption and a ring of debris.

Where tides come from:

Gravity weakens with distance, so the near and far sides of the moon feel different pulls. This difference stretches the moon along the axis toward the planet. It grows as $1/r^3$:

— twice as close means eight times stronger disruption. That is why close-in moons and comets are so vulnerable to tides.

The Roche limit and planetary rings:

Below the Roche limit the tidal force overcomes an object’s own gravity and tears it into debris that spreads into a ring — this is how Saturn’s rings formed, among others. The limit depends on the densities of the planet and the moon:

$R_p$ — planet radius, $\rho_p,\rho_k$ — densities of the planet and the moon. In the chart on the right the $1/r^3$ curve crosses the self-gravity level exactly at $d_{Roche}$ — to the left of it the moon cannot survive.

Why this matters for the exam:

The relation $F\propto 1/r^3$ explains tides, the disruption of moons and the origin of rings — and the exponent 3 (not 2 as in the law of gravity) is a classic trap in problems. The trick: compare r with $d_{Roche}$ and check whether the tides beat its own gravity.

Remember: the tidal force is the difference in gravity between an object’s sides and grows as 1/r³; below the Roche limit it tears a moon into a ring.